Name:
CSU ID:
Homework 1
August 31, 2015
Show work to receive credit. If answers can be defined as rational
numbers then they should be.
1. Find the point of intersection of the planes 4x−2y +z = 0, x−3y +z =
5, and 6x + 2y − 11z = 7.
2. Find the point of intersection of the planes 5x + 8y + z − 3w = 12,
2x + 7y + 6z + w = 0, −2x + 3y + 4z + 8w = 8, and 7x + z + w = 6.
3. Find the line of intersection of the planes 4x−2y+z = 0, x−3y+z = 5,
and 3x + y = −5. The solution should be expressed in vector format
with a parameter.
4. (a) Express in augmented matrix form the following system of equations. (You may have already used this method on the first two problems.) (b) Determine if a solution exists, and if so, express it in vector
form.
4x1 − 2x2 + x3 + 4x4 − 3x5 = 1
2x1 − 4x2 + x3 + 3x4 + 6x5 = 1
−x1 + 9x2 + 4x3 + 2x4 + 3x5 = 0
(1)
5. WITHOUT using determinants, verify that if ad − bc 6= 0, then the
system of equations
ax1 + bx2 = b1
cx1 + dx2 = b2
has a unique solution. You are expected to know this throughout the
semester.
6. (a) Find the parabola of the form y = ax2 + bx + c given that it
passes through the points (−1, 2), (0, 0), and (1, −7). It should
be solved using matrices.
(b) Find the parabolas vertex. Hint: Use a basic fact from first
semester calculus.
7. Give restrictions on a,b, and c such that the following linear system is
consistent.
x − y + 2z = a
2x + 4y − 3z = b
4x + 2y + z = c
8. Consider the system
5x1 − 2x2 + 3x3 + x4 = 34
−10x1 + 4x2 − 5x3 − 5x4 + 2x5 = −82
15x1 − 6x2 + 14x3 − 12x4 + 16x5 = 50
(a) Identify A, ~x, ~b corresponding to the system’s matrix format.
(b) Define the augmented matrix associated with the system.
(c) Bring the augmented matrix to row reduced echelon form. Use a
calculator and express the solution in terms of fractions.
(d) Define the solution of the system in terms of the free variable x4
and x5 . The solution MUST BE given in vector format.
(e) Define the solution of the system in term sof the free variables x3
and x4 . The solution MUST BE given in vector format. (Easiest
approach is to reorder the columns of the matrix.)
(f) Find the LU decomposition of A.